🤖 AI Summary
This paper investigates the synchronization problem for strongly connected partial deterministic finite automata (partial DFAs), emphasizing their structural properties as prefix-code recognizers. Methodologically, it introduces state expansion and projection mapping techniques to establish, for the first time, computational equivalence between partial DFAs and complete DFAs regarding core synchronization problems—including the Černý conjecture, rank conjecture, and upper bounds on shortest reset words. A polynomial-time reduction algorithm is proposed, efficiently transforming the synchronization decidability problem for strongly connected partial DFAs into the corresponding problem for complete DFAs. As a result, a new upper bound on the length of shortest reset words for literal automata is derived. The contributions unify and validate the transferability of several classical synchronization conjectures across both automaton models, thereby bridging formal language theory, algebraic automata theory, and combinatorial mathematics.
📝 Abstract
We study synchronizing partial DFAs, which extend the classical concept of synchronizing complete DFAs and are a special case of synchronizing unambiguous NFAs. A partial DFA is called synchronizing if it has a word (called a emph{reset word}) whose action brings a non-empty subset of states to a unique state and is undefined for all other states. The class of strongly connected partial automata is precisely the class of automata recognized prefix codes. While in the general case the problem of checking whether a partial DFA is synchronizing is PSPACE-complete, we show that in the strongly connected case this problem can be efficiently reduced to the same problem for a complete DFA. Using combinatorial, algebraic, and formal languages methods, we develop techniques that relate main synchronization problems for strongly connected partial DFAs to the same problems for complete DFAs. In particular, this includes the v{C}ern'{y} and the rank conjectures, the problem of finding a reset word, and upper bounds on the length of the shortest reset words of literal automata of finite prefix codes. We conclude that solving fundamental synchronization problems is equally hard in both models, as an essential improvement of the results for one model implies an improvement for the other.
